# Vessel theory: RAO quality checks

RAOs (particularly the phases) are difficult, abstract concepts which makes them difficult to check. It is extremely important to check them, since the same difficulty applies to the people who derived the data in the first place. RAOs and phases, even from the most respected sources, are notoriously error-prone!

Fortunately, there are a few natural points of reference where we know what must be going on. The most obvious and useful ones are responses at very short and very long wave periods.

## Displacement RAOs

In very short period waves, the vessel inertia suppresses response, so for all degrees of freedom the expected displacement RAO amplitude is zero (and phase is then irrelevant).

In very long waves (typically wave periods over 20 seconds for ships or 30 seconds for semisubmersibles) the vessel will move like a raft on the wave surface. The tables below give the expected displacement RAO amplitudes and phase lags for a free-floating vessel in very long waves.

### Expected RAOs for a vessel in very long waves

(180°)
From astern
(0°)
From port
(270°)
From starboard
(90°)
Amp. Phase lag Amp. Phase lag Amp. Phase lag Amp. Phase lag
Surge 1 -90° 1 +90° 0 ~ 0 ~
Sway 0 ~ 0 ~ 1 -90° 1 +90°
Heave 1 1 1 1
Roll 0 ~ 0 ~ 1 -90° 1 +90°
Pitch 1 +90° 1 -90° 0 ~ 0 ~
Yaw 0 ~ 0 ~ 0 ~ 0 ~

\begin{array}{lll} &\textbf{Towards direction $\beta$} \\ &\textbf{Amplitude} & \textbf{Phase Lag} \\ \textbf{Surge} & \lvert\cos\beta\rvert & \begin{aligned} +90\degree \text{ if }\cos\beta\gt0 \\ -90\degree \text{ if }\cos\beta\lt0 \end{aligned} \\ \textbf{Sway} & \lvert\sin\beta\rvert & \begin{aligned} +90\degree \text{ if }\sin\beta\gt0 \\ -90\degree \text{ if }\sin\beta\lt0 \end{aligned} \\ \textbf{Heave} & 1 & 0\degree \text{ for all }\beta \\ \textbf{Roll} & \lvert\sin\beta\rvert & \begin{aligned} +90\degree \text{ if }\sin\beta\gt0 \\ -90\degree \text{ if }\sin\beta\lt0 \end{aligned} \\ \textbf{Pitch} & \lvert\cos\beta\rvert & \begin{aligned} -90\degree \text{ if }\cos\beta\gt0 \\ +90\degree \text{ if }\cos\beta\lt0 \end{aligned} \\ \textbf{Yaw} & \lvert\frac12\sin(2\beta)\rvert & \begin{aligned} 180\degree \text{ if }\sin(2\beta)\gt0 \\ 0\degree \text{ if }\sin(2\beta)\lt0 \end{aligned} \end{array} See direction conventions for details of the way in which the direction $\beta$ is interpreted in OrcaFlex.

 Warning: The expected yaw RAOs given in the above table only apply to slender vessels whose displacement is distributed along the vessel x-axis.
 Notes: In these tables, the translational amplitudes are non-dimensionalised against wave amplitude and the rotational amplitudes are non-dimensionalised against maximum wave slope. The phases given are lags relative to the wave crest, so that +90° means that the maximum positive motion occurs 90° after the wave crest passes the vessel. In these tables we use the conventions of positive surge is forward, positive sway is to port, positive heave is up, positive roll is starboard down, positive pitch is bow down and positive yaw is bow to port. When the amplitude is zero the phase value is irrelevant; this is indicated in the tables by '~'.

You can check RAOs in a couple of simple ways. OrcaFlex provides RAO graphs that can identify some types of errors. In addition to this, it is often valuable to run quick simulations with only the vessel in the model and then check that the motions you see are sensible.

Consider a ship in waves coming from ahead. Set up a simple OrcaFlex model with the vessel only – nothing else – set the vessel's primary motion to none, superimposed motion to RAOs + harmonics, and run a short simulation (say 10 seconds build-up plus 2 wave periods). Use a large wave height (20m) and long time step, say 0.1 seconds. When the run is finished (almost instantaneous for such a trivial case), replay the last wave period and check that the motion of the ship is realistic. The best view direction is horizontal, normal to the direction of travel of the waves. With the waves coming from the right on screen, then in the wave crest the ship should be at maximum heave up and moving to the left; in the trough you should observe the opposite, so the ship should be at its lowest height and moving to the right. At the point of maximum wave slope as the crest approaches, the ship should be at maximum surge forwards into the wave and maximum pitch angle with the bow up. If the phase convention has been misunderstood (e.g. leads have been read as lags) then the motion will be obviously wrong and you should go back and re-examine the data, or check your interpretation with the data source.

This is an excellent check for phases, which are usually the most troublesome to get right. It is not quite so good for amplitudes, but it is nevertheless worth pursuing. If the wave is very long compared to the ship, then the ship should move like a small particle in the water surface. Heave amplitude should be equal to wave amplitude and pitch motion should keep the deck of the ship parallel to the water surface. Surge amplitude should also be equal to wave amplitude in deep water, but will be greater in shallow water in which the wave particle orbits are elliptical.

The check can be extended to other wave directions. Broadly speaking, we may expect the motion to be predominantly in the wave direction, with the phasing of surge and sway such that the components in the wave direction reinforce each other. Similarly, roll and pitch phasing should be such that the components of rotation about an axis normal to the wave direction reinforce each other. Yaw phasing for a ship in seas off the bow should be such that the ship yaws towards the broadside on position as the wave crest passes: this is easiest to see in a near-plan view. Generally speaking, if it looks right in long waves, it probably is right. If not, then think again!